Well, in the first line you must see that your apartment owner registers you upon arrival within the first 5 business days - you've been around and you are probably aware of that. You would also know about the local medical insurance that you will have to buy upon landing just for the sake of passing the passports control (unless you are on a British passport).
Weather. Well, it's autumn now and it is getting colder every day. Something like +15C at daytime and +8C at nighttime. From my experience BBC weather forecast is rather accurate.
P.S. By the way, there is now a currency crisis in Belarus. The hard currency such as USD or EUR is impossible to obtain from the banks and therefore the black market rates are substantially higher than those at the banks who are only buying it. With this in mind you should exchange your dollars or euro step by step to make sure you don't have a load of these by the time you have to leave. The black market dealers (or simply desperate business people who need to have their business going) prefer US dollars rather than euro. Since most ATMs in Minsk do not produce hard currency it will be good to bring some cash.
Since US dollar is losing value, the prices for consumer goods are growing more expensive almost every day. And only those who earn in hard currency or carry it around are in a very beneficial position.
Andrei